The connection between topology and computer science is based on two fundamental insights: the first, which can be traced back to the beginning ofrecursion theory, and even intuitionism, is that computable functions are necessarily continuous when input and output domains are equipped with their natural topologies. The second, due to M. B. Smyth in 1981, is that the observable properties of computational domains are contained in the collection of open sets. The first insight underlies Dana Scott's cate}ories ofsemantics domains, which are certain topological spaces with continuous functions. The second insight was made fruitful for computer science by Samson Abramsky, who showed in his 'Domain Theory in Logical Fonn' that instead ofworking with Scott's domains one can equivalently work with lattices of observable properties. Thus he established a precise link between denotational semantics and program logic. MathematicaIly, the framework for Abramsky's approach is that of Stone duality, which in general tenns studies the relationship between topological spaces and their lattices of opens sets. While for his purposes, Abramsky could rely· on existing duality results established by Stone in 1937, it soon became clear that in order to capture continuous domains, the duality had to be extended. Continuous domains are of interest to semantics because of the need to model the probabilistic behavior and computation over real numbers. The extension of the Stone duality was achieved by Jung and Siinderhaufin 1996; the main outcome of this investigation is the realization that the observable properties of a continuous space fonn a strong proximity lattice. The present thesis examines strong proximity lattices with the tools of Priestley duality, which was introduced in 1970 as an alternative to Stone's duality for distributive lattices. The advantage of Priestley duality is that it yields compact Hausdorff spaces and thus stays within classical topological ideas. The thesis shows that Priestley duality can indeed be extended to cover strong proximity lattices, and identifies the additional structure on Priestley spaces that corresponds to·the proximity relation. At least three different types of morphism have been defmed between strong proximity lattices, and the thesis shows that each of them can be used in Priestley duality. The resulting maps between Priestley spaces are characterized and given a computational interpretation. . This being an alternative to the Jung-Siinderhauf duality, it is examined how the two dualities are related on the side of topological spaces. FinaIly, strong proximity lattices can be seen as algebras ofthe logic MLS, introduced by Jung, Kegelmann, and Moshier. The thesis examines how the central notions ofMLS are transfonned by Priestley duality.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:487219 |
| Date | January 2006 |
| Creators | El-Zawawy, Mohamed Abdel-Moneim Mahmoud Mohamed |
| Publisher | University of Birmingham |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
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